Constant Multiple Rule Integration Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Constant Multiple Rule Integration Calculator helps you integrate a constant times a function such as 4x^3 or 2sin(x). A constant factor can be moved outside the integral, the remaining function is integrated, and the constant multiplies back in at the end.

Step-by-step method

  1. Identify the constant multiple k and the function f(x).
  2. Write the constant multiple rule for integration.
  3. Pull the constant outside the integral.
  4. Integrate the remaining function, multiply back, and add C.

Formula:

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Example 1:

\(\int 4 x^{3}\, dx\)

Step 1 - Identify the constant multiple k and the function f(x).

In this problem: The constant is \(k = 4\) and the function is \(x^{3}\).

\(\int 4 x^{3}\, dx\)

Step 2 - Write the constant multiple rule for integration.

In this problem: A constant factor can be moved outside the integral.

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Step 3 - Pull the constant outside the integral.

In this problem: Move \(4\) in front of the integral sign.

\(\int 4 x^{3}\, dx = 4\int x^{3}\, dx\)

Step 4 - Integrate the remaining function, multiply back, and add C.

In this problem: The integral of \(x^{3}\) is \(\frac{x^{4}}{4}\). Multiplying by \(4\) gives \(x^{4}\).

\(4\int x^{3}\, dx = 4 \cdot \left(\frac{x^{4}}{4}\right) = x^{4} + C\)

Final answer:

\(\int 4 x^{3}\, dx = x^{4} + C\)

Example 2:

\(\int 2 \sin{\left(x \right)}\, dx\)

Step 1 - Identify the constant multiple k and the function f(x).

In this problem: The constant is \(k = 2\) and the function is \(\sin{\left(x \right)}\).

\(\int 2 \sin{\left(x \right)}\, dx\)

Step 2 - Write the constant multiple rule for integration.

In this problem: A constant factor can be moved outside the integral.

\(\int k \cdot f\left(x\right)\, dx = k\int f\left(x\right)\, dx\)

Step 3 - Pull the constant outside the integral.

In this problem: Move \(2\) in front of the integral sign.

\(\int 2 \sin{\left(x \right)}\, dx = 2\int \sin{\left(x \right)}\, dx\)

Step 4 - Integrate the remaining function, multiply back, and add C.

In this problem: The integral of \(\sin{\left(x \right)}\) is \(- \cos{\left(x \right)}\). Multiplying by \(2\) gives \(- 2 \cos{\left(x \right)}\).

\(2\int \sin{\left(x \right)}\, dx = 2 \cdot \left(- \cos{\left(x \right)}\right) = - 2 \cos{\left(x \right)} + C\)

Final answer:

\(\int 2 \sin{\left(x \right)}\, dx = - 2 \cos{\left(x \right)} + C\)